Thermodynamics

Chapter 13 | 697Discussion This result is about 6 percent greater than the result obtained inpart (b) by using Kay’s rule. But it is more than twice the result obtained byassuming the mixture to be an ideal gas.TOPIC OF SPECIAL INTEREST*Chemical Potential and the Separation Work of MixturesWhen two gases or two miscible liquids are brought into contact, they mixand form a homogeneous mixture or solution without requiring any workinput. That is, the natural tendency of miscible substances brought into contactis to mix with each other. As such, these are irreversible processes, andthus it is impossible for the reverse process of separation to occur spontaneously.For example, pure nitrogen and oxygen gases readily mix whenbrought into contact, but a mixture of nitrogen and oxygen (such as air)never separates into pure nitrogen and oxygen when left unattended.Mixing and separation processes are commonly used in practice. Separationprocesses require a work (or, more generally, exergy) input, and minimizingthis required work input is an important part of the design process ofseparation plants. The presence of dissimilar molecules in a mixture affecteach other, and therefore the influence of composition on the properties mustbe taken into consideration in any thermodynamic analysis. In this sectionwe analyze the general mixing processes, with particular emphasis on idealsolutions, and determine the entropy generation and exergy destruction. Wethen consider the reverse process of separation, and determine the minimum(or reversible) work input needed for separation.The specific Gibbs function (or Gibbs free energy) g is defined as thecombination property g h Ts. Using the relation dh v dP Tds, thedifferential change of the Gibbs function of a pure substance is obtained bydifferentiation to bedg v dP s dTordG V dP S dT1pure substance2(13–27)For a mixture, the total Gibbs function is a function of two independentintensive properties as well as the composition, and thus it can be expressedas G G(P, T, N 1 , N 2 , . . . , N i ). Its differential isdG a 0G0P b dP a 0GT,N 0T b dT aiP,Na 0G0N ibP,T,N jdN i 1mixture2(13–28)where the subscript N j indicates that the mole numbers of all components inthe mixture other than component i are to be held constant duringdifferentiation. For a pure substance, the last term drops out since the compositionis fixed, and the equation above must reduce to the one for a puresubstance. Comparing Eqs. 13–27 and 13–28 givesdG V dP S dT aim i dN i ordg v dP s dT aim i dy i(13–29)*This section can be skipped without a loss in continuity.